There is no reason to support a minimum wage if you don't care about quality of life.
There are also others way to create high unemployment: making it very risky and expensive to get rid of mishires or to reduce headcount if the company isn't going so well. Making management more difficult once the headcount reaches a certain level (because of unions or the creation of employee boards mandated by law) -- that leads to companies trying to stay below such ceilings. Others would be salaries/pensions/benefits tied to how long one has worked for the same company -- which discourages moves to other companies (that might fit ones personality or talents better -- or that has better management). The housing market also has an influence: there might be opportunities in a different city but how easy is it to move? It is relatively easy if you rent your house or appartment. Not so easy if you own it, especially if selling it means you realize a loss. Lots of laws and regulations in a country influence the structure of the housing market (how liquid it is, what loan types are available, how many people rent instead of owning, etc.).
Sweden has a quite high unemloyment, especially for young people, and Sweden has no minimum wage.
No one is going to work for much less than they'll get from unemployment, and just try starting a company paying less than the union collective bargaining rates and see how long it takes before a bunch of thugs harass you out of business.